Fixed point spaces, primitive character degrees and conjugacy class sizes

Let $G$ be a finite group that acts on a nonzero finite dimensional vector space $V$ over an arbitrary field. Assume that $V$ is completely reducible as a $G$-module, and that $G$ fixes no nonzero vector of $V$. We show that some element $g\in G$ has a small fixed-point space in $V$. Specifically, we prove that we can choose $g$ so that $\dim \mathbf{C}_V(g)\le(1/p)\dim V$, where $p$ is the smallest prime divisor of $\vert G\vert$.